Circular Chromatic Ramsey Number

Let χc(H) denote the circular chromatic number of a graph H. For graphs F and G, the circular chromatic Ramsey number Rχc(F,G) is the infimum of χc(H) over graphs H such that every red/blue edge-coloring of H contains a red copy of F or a blue copy of G. We characterize Rχc(F,G) in terms of a Ramsey problem for the families of homomorphic images of F and G. Letting zk = 3 − 2 −k, we prove that zk−1 ≤ χc(G) ≤ zk implies 2zk−1 ≤ Rχc(G,G) ≤ 2zk. For integer k with k > 1, there is a graph G with χc(G) ≥ k and Rχc(G,G) ≤ k(k − 1). Our most difficult result is Rχc(F,G) = 4 when χc(F ), χc(G) ∈ (2, 5 2 ]. As a consequence, no graph G satisfies 4 < Rχc(G,G) < 5. We also prove 14 3 ≤ Rχc(C3, C5) ≤ 5 and 4 ≤ Rχc(C3, C7) ≤ 9 2 .